Determinants and Solutions of Linear Systems of Equations

1 Introduction

In this paper, we will study de terminants and solutions of linear systems of
equations in some detail. We will learn the basics for each and expand on them .

2 Determinants

A determinant is a mathematical object which is very useful in the analysis and
solution of systems of linear equations . Determinants are only defined for square
matrices. A square matrix has horizontal and vertical dimensions that are the
same (i.e., an nxn matrix). The difference between the form of a matrix and a
determinant of a matrix is that a determinant is displayed using straight lines
in-place of the square brackets . The determinant is a scalar quantity, which
means a one-comp onent quantity . The determinant is most often used to:

• test whether or not a matrix has an inverse
• test for linear dependence of vectors (in certain situations)
• test for existence/uniqueness of solutions of linear systems of equations

3 An nxn Matrix

In an nxn matrix, we fol low certain rules for the appearance of the matrix.
We let symbolize the i^th row. . If this row were to be multiplied
by α, the the row would appear to be . Also, if
two rows were added, the i^th row and the j^th column, we would have
. A unit matrix appears with the following rows:
(1,0,0,...,0),(0,1,0,...,0),...,(0,0,0,...,1). Also, the letters e₁, ..., e_n describe a unit
row.

In an nxn matrix there are a few forms in which a determinant is recognized.
First of all the determinant symbolizes the function of the n² variables
1, 2, ..., n). The determinant for this function can be written as:

4 Properties of Determinants

1.   (Invariance)
2. (Homogeneity)
3. (Normalization)

5 Rules for Determinants

Determinants are either written as |A| or detA. Now say that we delete the
i^th row and the j^th column a (n − 1)x(n − 1) submatrix A_ij is formed. The
determinant of this submatrix is the minor element of a_ij . The co factor of a _ij
is . We also write the cofactor as   . The following are some rules
for determinants.

a. |A| = |A'|, A' = transpose of 

b. If two rows (or columns) of A are interchanged, producing a matrix A₁,
then |A| = −|A|

c. If two rows (or columns) of A are identical, then —A—=0

d. If a row (or column), v, of A is replaced by kv producing a matrix A₁,
then |A|=k |A|.

e. If a scalar multiple kv , of the row (or column) is added to the row
(or column) , (i ≠ j) and the matrix A₁ results, then |A| = |A₁|.

f. A determinant may be evaluated in terms of cofactors: |A| =

6 2X2 Matrix

The most basic determinant is found using a 2x2 matrix in the form

The determinant of a 2x2 matrix is found using the following formula:

7 Example 1

2x2 Matrix Using the matrix

the determinant would be

8 3x3 Matrix

The next kind of matrix studied is the 3x3 matrix. The form of this matrix is

The determinant of a 3x3 matrix is found using the following formula:

9 Example 2

3x3 Matrix Using the matrix

the determinant would be

10 Solution of Linear Systems of Equations

Consider the system of n linear equations in n unknowns

11 Cramer’s Rule

The above formula is Cramer ’s Rule. It states that if then

possesses a unique solution given by

Cramer’s rule is used to solve a set of n linear equations in n unknowns. It uses
determinants to obtain the solution.

12 The Alternative Theorem

This formula that results from Cramer’s Rule is the Alternative Theorem. Also
described as the homogeneous system

possesses a non-trivial solution (i.e., a solution other than
if and only if |A|=0. If for a fixed there are solutions to the
non-homogeneous system

for every selection of the quantities bi, then |A| ≠ 0 and the homogeneous system
had only the trivial solution.